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Essential Questions. How do we explain that probability can be used to help determine if good decisions are made? How do we use probabilities to analyze decisions and strategies?.
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Essential Questions • How do we explain that probability can be used to help determine if good decisions are made? • How do we use probabilities to analyze decisions and strategies?
In experiments with numerical outcomes, the expected value(EV) is the weighted average of the numerical outcomes of a probability experiment.
Example 1: Finding Expected Value 1 1 1 1 1 1 The sides of a six-sided number cube are labeled 1, 1, 3, 3, 9, and 9. E(V) = 1 + + 6 + 3 1 3 9 9 + + 6 6 6 6 6 a. What is the expected value of the number cube? 1 1 3 3 9 9 1 1 1 1 1 1 6 6 6 6 6 6
Example 1: Continued b. What is the expected value of rolling two number cubes, one labeled as described in part A and the other labeled 1– 6?
Example 2: Finding Expected Value What is the expected value of rolling the six sided number cube as shown in the net? 1 2 2 3 3 5 1 1 1 1 1 1 6 6 6 6 6 6
Example 3 : Expected Value in Real-World Situations On a mountain, it takes Sam 2 hours to climb the southern route, unless there is ice, which increases the time to 4 hours. It takes him 2.5 hours to climb the eastern route, unless there is ice, which increases the time to 3 hours. If the chance of ice is 20% on the southern route and 40% on the eastern route, which route should Sam take if he wants to finish the climb as soon as possible? He should take the southern route.
Example 4: Expected Value in Real-World Situations Jack can take one of three routes to work each day. Route A takes 16 minutes, Route B takes 10 minutes, and Route C takes 20 minutes. There is a 40% chance he will encounter an accident in Route A, which increases travel time to 25 minutes. There is also a 20% chance he will encounter a traffic jam if he takes Route B, which increases his travel time to 40 minutes. He has a 10% chance of experiencing a delay in Route C, which increases his travel time to 32 minutes. Which route should Jack take to work each day? He should take Route B.
Example 5: The Monty Hall Problem In a TV game show, a car key is hidden in one of five bags. The other bags contain fake keys. Once the contestant picks a bag, the host, knowing where the key is located, opens a bag with a fake key. As the contestant answers questions correctly, he continues to open bags with fake keys until two bags remain: one with the car key and one with a fake key. At this time, he offers the contestant a chance to switch bags. Find the expected value of sticking with the original bag and the expected value of switching bags.
Example 6: The Financial Aid Decision Mikayla is applying to 3 colleges. She makes estimates of her chances of being accepted, and estimates of her chances of receiving financial aid from each, presented in table: At which college is she most likely to be both accepted and receive financial aid? She has a higher probability of being accepted in College C with financial aid.
